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We denote the image of a group element <math>g</math> under a function <math>\omega</math> with <math>g^\omega</math>. The distributivity can then be expressed as
:<math>(g \circ h)^{\omega} = g^{\omega} \circ h^{\omega} \quad \forall \omega \in \Omega, \forall g,h \in G.</math>
Using [[category theory]], a '''group with operators''' can be defined as an object of a [[functor category]] '''Grp'''<sup>'''M'''</sup> where '''M''' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the category of groups. This definition is equivalent to the previous one.
A [[subgroup]] <math>S</math> of <math>G</math> is called '''stable subgroup''', <math>\Omega</math>-'''subgroup''' or <math>\Omega</math> '''invariant subgroup''' if it respects the hometheties, that is
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